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Chapter 4: Problem 16

Simplify each expression. $$-2\left(y^{3}+2 y^{2}-y\right)-3\left(3 y^{3}+y\right)$$

Short Answer

Expert verified
The simplified expression is \(-11y^{3}-4y^{2}-y\).

Step by step solution

01

Distribute Constants

Apply the distributive property of multiplication over addition to multiply each constant to all the terms inside the parentheses. The expression becomes \[ -2y^{3}-4y^{2}+2y-9y^{3}-3y \].
02

Combine Like Terms

Combine the like terms in the expression. That is, add or subtract terms that have the same variable to the same power. The combined expression becomes \[ -11y^{3}-4y^{2}-1y \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Simplification
Polynomial simplification is the process of making a polynomial expression less complicated or easier to manage. This involves reducing the expression to its simplest form. If you're new to polynomials, think of them as a mathematical sentence made up of variables and coefficients. A polynomial expression can have several terms with different variables and powers.

During the simplification, you can follow these steps:
  • Identify each term in the polynomial. A term can be a number, a variable like "y", or a combination of both like "y^3".
  • Simplify each term if possible. Look for shared factors or ways to reduce fractions.
  • Use properties like combining like terms or applying the distributive property to simplify further.
The goal is to have the expression in its simplest form, which makes it easier for calculations and further mathematical operations.
Distributive Property
The distributive property is an essential math rule that helps break down expressions into more workable parts. It's like distributing "gifts" into several "bags". When you distribute numbers over a set of terms, you multiply the number outside the parentheses by each term within the parentheses.

For example, consider the expression \(-2(y^{3}+2y^{2}-y)\). With the distributive property, you multiply \-2\ by \y^{3}\, \2y^{2}\, and \-y\. Each term becomes:
  • \(-2 \times y^{3} = -2y^{3}\)
  • \(-2 \times 2y^{2} = -4y^{2}\)
  • \(-2 \times (-y) = 2y\)
You can apply the distributive property to any polynomial to break it down for further simplification, making the process easier to manage.
Combining Like Terms
Combining like terms is a crucial step in simplifying polynomial expressions. This step makes sure that an expression is as compact as possible by grouping and reducing terms that are "alike."

When we say "like terms," we refer to terms in a polynomial that have the same variables raised to the same power. For instance, in the expression \(-11y^{3} - 4y^{2} - 1y\), the like terms would be those with the same degree in "y".
  • The term \(-11y^{3}\) is a like term with any other \y^{3}\ term.
  • The term \(-4y^{2}\) stands alone here since there are no other \y^{2}\ terms.
  • The term \(-1y\) is a like term with any other \y\ term.
By grouping the terms with similar variables and powers, you reduce the complexity of the expression. This process helps in solving equations more efficiently, as well as keeping expressions tidy and easier to read.

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Most popular questions from this chapter

Simplify or solve as appropriate. $$(2 s-3)(s+2)=(2 s+1)(s-3)$$

If \(f(x)=x^{2}-2 x+3,\) find each value. $$f(1.2)$$

Simplify or solve as appropriate. $$4+(2 y-3)^{2}=(2 y-1)(2 y+3)$$

Some students threw balloons filled with water from a dormitory window. The height \(h\) (in feet) of the balloons \(t\) seconds after being thrown is given by the polynomial function $$h=f(t)=-16 t^{2}+12 t+20$$ How far above the ground is a balloon 1.5 seconds after being thrown?

Give the degree of each polynomial. $$17^{2} x$$
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